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ISRN Geometry
Volume 2012 (2012), Article ID 251213, 16 pages
http://dx.doi.org/10.5402/2012/251213
Research Article

Hemi-Slant Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India

Received 25 April 2012; Accepted 25 June 2012

Academic Editors: T. Friedrich and M. Khalkhali

Copyright © 2012 S. M. Khursheed Haider et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. We give an example of hemi-slant lightlike submanifold and establish two characterization theorems for the existence of such submanifolds. We prove some theorems which ensure the existence of minimal hemi-slant lightlike submanifolds and obtain a condition under which the induced connection on M is a metric connection. An example of proper minimal hemi-slant lightlike submanifolds is also given.

1. Introduction

Given a semi-Riemannian manifold, one can consider its lightlike submanifold whose study is important from application point of view and difficult in the sense that the intersection of normal vector bundle and tangent bundle of these submanifolds is nonempty. This unique feature makes the study of lightlike submanifolds different from the study of nondegenerate submanifolds. The general theory of lightlike submanifolds was developed by Kupeli [1]; Duggal and Bejancu [2]. On the other hand, the concepts of transversal and screen transversal lightlike submanifolds of an indefinite Kaehler manifold were given by Sahin [3, 4]. In Sasakian setting, these submanifolds were investigated by Yildirim and Sahin [57].

In the present paper, we introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. The paper is arranged as follows. In Section 2, we recall definitions for indefinite Kenmotsu manifolds and give basic information on the lightlike geometry needed for this paper. In Section 3, after defining hemi-slant lightlike submanifolds immersed in an indefinite Kenmotsu manifold, we give an example which ensures the existence of hemi-slant lightlike submanifolds. We obtain integrability conditions of distributions involved, investigate the geometry of leaves of distributions, and establish two characterization theorems for the existence of hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold. We prove that there does not exist curvature invariant hemi-slant lightlike submanifold in an indefinite Kenmotsu space form with some condition on 𝑐 and obtain a geometric condition under which the induced connection on 𝑀 is a metric connection. In Section 4, we study minimal hemi-slant lightlike submanifolds and give an example of proper minimal hemi-slant lightlike submanifolds immersed in 𝑅92.

2. Preliminaries

An odd dimensional semi-Riemannian manifold (𝑀,𝑔) is said to be an indefinite contact metric manifold [8] if there exists a (1,1) tensor field 𝜙, a vector field 𝑉, called the characteristic vector field, and its 1-form 𝜂 satisfying 𝑔(𝜙𝑋,𝜙𝑌)=𝑔(𝑋,𝑌)𝜖𝜂(𝑋)𝜂(𝑌),𝜙𝑔(𝑉,𝑉)=𝜖2𝑋=𝑋+𝜂(𝑋)𝑉,𝑔(𝑋,𝑉)=𝜖𝜂(𝑋),𝑋,𝑌Γ(𝑇𝑀),(2.1) where 𝜖=±1. It follows that 𝜙𝑉=0,𝜂𝑜𝜙=0,𝜂(𝑉)=𝜖.

An indefinite almost contact metric manifold 𝑀 is said to be an indefinite Kenmotsu manifold [9] if 𝑋𝑉=𝑋+𝜂(𝑋)𝑉,(2.2)𝑋𝜙𝑌=𝑔(𝜙𝑋,𝑌)𝑉+𝜖𝜂(𝑌)𝜙𝑋.(2.3) for any 𝑋,𝑌Γ(𝑇𝑀).

We follow [2] for the notation and formulae used in this paper. A submanifold (𝑀𝑚,𝑔) immersed in a semi-Riemannian manifold (𝑀𝑚+𝑛,𝑔) is called a lightlike submanifold if the metric 𝑔 induced from 𝑔 is degenerate and the radical distribution Rad𝑇𝑀 is of rank 𝑟, where 1𝑟𝑚. Let 𝑆(𝑇𝑀) be a screen distribution which is a semi-Riemannian complementary distribution of Rad𝑇𝑀 in 𝑇𝑀, that is, 𝑇𝑀=Rad𝑇𝑀𝑆(𝑇𝑀).(2.4) Consider a screen transversal vector bundle 𝑆(𝑇𝑀), which is a semi-Riemannian complementry vector bundle of Rad𝑇𝑀 in 𝑇𝑀. Since for any local basis {𝜉𝑖} of Rad𝑇𝑀, there exist a local null frame {𝑁𝑖} of sections with values in the orthogonal complement of 𝑆(𝑇𝑀) in [𝑆(𝑇𝑀)] such that 𝑔(𝜉𝑖,𝑁𝑗)=𝛿𝑖𝑗, it follows that there exist a lightlike transversal vector bundle ltr(𝑇𝑀) locally spanned by {𝑁𝑖} [2, page 144]. Let tr(𝑇𝑀) be complementary (but not orthogonal) vector bundle to 𝑇𝑀 in 𝑇𝑀|𝑀. Then tr(𝑇𝑀)=ltr(𝑇𝑀)𝑆𝑇𝑀.𝑇𝑀||𝑀[]=𝑆(𝑇𝑀)(Rad𝑇𝑀)ltr(𝑇𝑀)𝑆𝑇𝑀.(2.5) The Gauss and Weingarten equations are as follows: 𝑋𝑌=𝑋𝑌+(𝑋,𝑌),𝑋,𝑌Γ(𝑇𝑀),𝑋𝑈=𝐴𝑈𝑋+𝑡𝑋𝑈,𝑋Γ(𝑇𝑀),𝑈Γ(tr(𝑇𝑀)),(2.6) where {𝑋𝑌,A𝑈𝑋} and {(𝑋,𝑌),𝑡𝑋𝑈} belong to Γ(𝑇𝑀) and Γ(tr(𝑇𝑀)), respectively, and 𝑡 are linear connections on 𝑀 and on the vector bundle tr(𝑇𝑀), respectively. Moreover, we have 𝑋𝑌=𝑋𝑌+𝑙(𝑋,𝑌)+𝑠(𝑋,𝑌),(2.7)𝑋𝑁=𝐴𝑁𝑋+𝑙𝑋𝑁+𝐷𝑠(𝑋,𝑁),(2.8)𝑋𝑊=𝐴𝑊𝑋+𝑠𝑋𝑊+𝐷𝑙(𝑋,𝑊),(2.9) for each 𝑋,𝑌Γ(𝑇𝑀), 𝑁Γ(ltr(𝑇𝑀)) and 𝑊Γ(𝑆(𝑇𝑀)). If we denote the projection of 𝑇𝑀 on 𝑆(𝑇𝑀) by 𝑃, then by using (2.7)–(2.9) and the fact that is a metric connection, we obtain 𝑔𝑠+(𝑋,𝑌),𝑊𝑔𝑌,𝐷𝑙𝐴(𝑋,𝑊)=𝑔𝑊,𝑋,𝑌(2.10)𝑔(𝐷𝑠(𝑋,𝑁),𝑊)=𝑔𝑁,𝐴𝑊𝑋.(2.11) From the decomposition of the tangent bundle of a lightlike submanifold, we have 𝑋𝑃𝑌=𝑋𝑃𝑌+(𝑋,𝑃𝑌),(2.12)𝑋𝜉=𝐴𝜉𝑋+𝑋𝑡𝜉,(2.13) for 𝑋,𝑌Γ(𝑇𝑀) and 𝜉Γ(Rad𝑇𝑀). By using the above equation, we get 𝑔𝑙𝐴(𝑋,𝑃𝑌),𝜉=𝑔𝜉𝑋,𝑃𝑌.(2.14) It is important to note that the induced connection on 𝑀 is not a metric connection whereas is a metric connection on 𝑆(𝑇𝑀).

We denote curvature tensors of 𝑀 and 𝑀 by 𝑅 and 𝑅, respectively. The Gauss equation for 𝑀 is given by 𝑅(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍+𝐴𝑙(𝑋,𝑍)𝑌𝐴𝑙(𝑌,𝑍)𝑋+𝐴𝑠(𝑋,𝑍)𝑌𝐴𝑠(𝑌,𝑍)𝑋+𝑋𝑙(𝑌,𝑍)𝑌𝑙(𝑋,𝑍)+𝐷𝑙𝑋,𝑠(𝑌,𝑍)𝐷𝑙𝑌,𝑠+(𝑋,𝑍)𝑋𝑠(𝑌,𝑍)𝑌𝑠(𝑋,𝑍)+𝐷𝑠𝑋,𝑙(𝑌,𝑍)𝐷𝑠𝑌,𝑙,(𝑋,𝑍)(2.15) for all 𝑋,𝑌,𝑍Γ(𝑇𝑀).

The curvature tensor 𝑅 of an indefinite Kenmotsu space form 𝑀(𝑐) is given by [9] as follows: 𝑅(𝑋,𝑌)𝑍=𝑐34𝑔(𝑌,𝑍)𝑋+𝑔(𝑋,𝑍)𝑌𝑐+14𝜂(𝑋)𝜂(𝑍)𝑌𝜂(𝑌)𝜂(𝑍)𝑋+𝑔(𝑋,𝑍)𝜂(𝑌)𝑉+𝑔(𝑌,𝑍)𝜂(𝑋)𝑉𝑔(𝜙𝑌,𝑍)𝜙𝑋𝑔(𝜙𝑋,𝑍)𝜙𝑌2,𝑔(𝜙𝑋,𝑌)𝜙𝑍(2.16) for any 𝑋,𝑌Γ(𝑇𝑀).

From now on, we denote (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(𝑇𝑀)) by 𝑀 in this paper.

3. Hemi-Slant Lightlike Submanifolds

The concept of pseudoslant (antislant) submanifolds with definite metric is given by Carriazo [10] which was renamed by Sahin as hemi-slant submanifolds and studied their warped product in Kaehler manifolds [11]. In this section, we introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold for which we need the following lemma from [3].

Lemma 3.1. Let 𝑀 be a 2𝑞-lightlike submanifold of an indefinite Kaehler manifold 𝑀 with constant index 2𝑞 such that 2𝑞<dim(𝑀). Then the screen distribution 𝑆(𝑇𝑀) of lightlike submanifold 𝑀 is Riemannian.

In view of the above lemma, we can define hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold as follows.

Definition 3.2. Let 𝑀 be a 2𝑞-lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2𝑞 with the structure vector field 𝑉 tangent to 𝑀 such that 2𝑞<dim𝑀. Then, the submanifold 𝑀 is said to be hemi-slant lightlike submanifold of 𝑀 if the following conditions are satisfied: (i)Rad𝑇𝑀 is a distribution on 𝑀 such that 𝜙(Rad𝑇𝑀)=ltr(𝑇𝑀);(3.1)(ii)for all 𝑥𝑈𝑀 and for each non zero vector field 𝑋 tangent to 𝑆(𝑇𝑀)=𝐷𝜃{𝑉} if 𝑋 and 𝑉 are linearly independent, then the angle 𝜃(𝑋) between 𝜙𝑋 and the vector space 𝑆(𝑇𝑀) is constant, where 𝐷𝜃 is the complementary distribution to 𝑉 in the screen distribution 𝑆(𝑇𝑀).

The angle 𝜃(𝑋) is called the slant angle of the distribution 𝐷𝜃. A hemi-slant lightlike submanifold is said to be proper if 𝐷𝜃0 and 𝜃0,𝜋/2.

From Definition 3.2, we have the following decomposition: 𝑇𝑀=Rad𝑇𝑀𝐷𝜃{𝑉},tr(𝑇𝑀)=ltr(𝑇𝑀)𝐹𝐷𝜃𝑇𝜇,𝑀=(Rad𝑇𝑀ltr(𝑇𝑀))𝐷𝜃𝐹𝐷𝜃𝜇{𝑉}.(3.2)

Denoting (𝑅𝑞2𝑚+1,𝜙𝑜,𝑉,𝜂,𝑔) as a manifold 𝑅𝑞2𝑚+1 with its usual Kenmotsu structure given by 𝜂=𝑑𝑧,𝑉=𝜕𝑍,𝑔=𝜂𝜂+𝑒2𝑧Σ𝑞/2𝑖=1𝑑𝑥𝑖𝑑𝑥𝑖+𝑑𝑦𝑖𝑑𝑦𝑖+Σ𝑚𝑖=𝑞+1𝑑𝑥𝑖𝑑𝑥𝑖+𝑑𝑦𝑖𝑑𝑦𝑖,𝜙𝑜Σ𝑚𝑖=1𝑋𝑖𝜕𝑥𝑖+𝑌𝑖𝜕𝑦𝑖+𝑍𝜕𝑧=Σ𝑚𝑖=1𝑌𝑖𝜕𝑥𝑖𝑋𝑖𝜕𝑦𝑖,(3.3) where (𝑥𝑖,𝑦𝑖,𝑧) are the cartesian coordinates.

The existence of proper hemi-slant lightlike submanifolds in indefinite Kenmotsu manifolds is ensured by the following example.

Example 3.3. Let 𝑀 be a submanifold of 𝑀=(𝑅92,𝑔) defined by 𝑥1=𝑠,𝑥2=𝑡,𝑥3=𝑢sin𝑤,𝑥4𝑦=sin𝑢,1=𝑡,𝑦2=𝑠,𝑦3=𝑢cos𝑤,𝑦4=cos𝑢,(3.4) where 𝑢,𝑤(0,𝜋/2) and 𝑅92 is a semi-Euclidean space of signature (,,+,+,+,+,+,+,+).
Then, a local frame of 𝑇𝑀 is given by 𝜉1=𝑒𝑧𝜕𝑥1+𝜕𝑦2,𝜉2=𝑒𝑧𝜕𝑥2+𝜕𝑦1,𝑋1=𝑒𝑧sin𝑤𝜕𝑥3+cos𝑢𝜕𝑥4+cos𝑤𝜕𝑦3sin𝑢𝜕𝑦4,𝑋2=𝑒𝑧𝑢cos𝑤𝜕𝑥3𝑢sin𝑤𝜕𝑦3,𝑉=𝜕𝑍.(3.5) We see that 𝑀 is a 2-lightlike submanifold with Rad𝑇𝑀=span{𝜉1,𝜉2} and 𝑆(𝑇𝑀)=span{𝑋1,𝑋2}{𝑉} which is Riemannian and can be easily proved that 𝑆(𝑇𝑀) is a slant distribution with slant angle 𝜃=𝜋/4. Moreover, the screen transversal bundle 𝑆(𝑇𝑀) is spanned by 𝑊1=𝑒𝑧sin𝑢𝜕𝑥4+cos𝑢𝜕𝑦4,𝑊2=𝑒𝑧sin𝑤𝜕𝑥3cos𝑢𝜕𝑥4+cos𝑤𝜕𝑦3+sin𝑢𝜕𝑦4,(3.6) and the transversal lightlike bundle ltr(𝑇𝑀) is spanned by 𝑁1𝑒=𝑧2𝜕𝑥1𝜕𝑦2,𝑁2=𝑒𝑧2𝜕𝑥2𝜕𝑦1.(3.7) It is not difficult to see that 𝜙𝜉1=𝑁2,𝜙𝜉2=𝑁1.(3.8) Hence, 𝑀 is a hemi-slant lightlike submanifold of 𝑅92.

For any 𝑋Γ(𝑇𝑀), we write 𝜙𝑋=𝑇𝑋+𝐹𝑋,(3.9) where 𝑇𝑋 and 𝐹𝑋 are the tangential and transversal components of 𝜙𝑋, respectively. Similarly, for 𝑊Γ(tr(𝑇𝑀)), we write 𝜙𝑊=𝐵𝑊+𝐶𝑊,(3.10) where 𝐵𝑊 and 𝐶𝑊 are the tangential and transversal components of 𝜙𝑊, respectively. If the projections on the distributions 𝑆(𝑇𝑀)=𝐷𝜃{𝑉} and Rad𝑇𝑀 are denoted by 𝑄1 and 𝑄2, respectively, then any 𝑋 tangent to 𝑀 can be written as 𝑋=𝑄1𝑋+𝑄2𝑋.(3.11) We observe that 𝑄1𝑋=𝑄1𝑋+𝜂(𝑋)𝑉. By applying 𝜙 to (3.11) and using (3.9), we conclude that 𝜙𝑋=𝑇𝑄1𝑋+𝐹𝑄1𝑋+𝐹𝑄2𝑋,(3.12)𝜙𝑄2𝑋=𝐹𝑄2𝑋,𝑇𝑄2𝑋=0,𝑇𝑄1𝑋Γ(𝑆(𝑇𝑀)).(3.13)

The following two theorems are the characterizations of hemi-slant lightlike submanifolds which are similar to the characterization of slant submanifolds in indefinite Hermitian manifolds given by Sahin [12].

Theorem 3.4. The necessary and sufficient condition for a 2𝑞-lightlike submanifold 𝑀 of an indefinite Kenmotsu manifold 𝑀 of index 2𝑞 with structure vector field 𝑉 tangent to 𝑀 to be hemi-slant lightlike submanifold is that (i)𝜙(ltr(𝑇𝑀)) is a distribution on 𝑀; (ii)for any vector field 𝑋 tangent to 𝑀 linearly independent of structure vector field 𝑉, there exists a constant 𝜆[1,0] such that 𝑄1𝑇2𝑄1𝑋=𝜆𝑄1𝑋𝜂𝑄1𝑋𝑉,(3.14)where 𝜆=cos2𝜃 and 𝜃 is the slant angle of 𝑀.

Proof. Assume that 𝑀 is a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝜙(Rad𝑇𝑀)=ltr(𝑇𝑀), from which we have 𝜙(ltr(𝑇𝑀))=Rad(𝑇𝑀). Hence 𝜙(ltr(𝑇𝑀)) is a distribution on 𝑀. In order to prove (ii), we observe that, for any 𝑋Γ(𝑇𝑀) linearly independent of structure vector field 𝑉 and 𝑄1𝑋Γ(𝐷𝜃), one has 𝑄cos𝜃1𝑋=𝑄𝑔1𝑄𝑋,1𝑇2𝑄1𝑋||𝑄1𝑋||||𝑇𝑄1𝑋||.(3.15) But cos𝜃(𝑄1𝑋) can also be computed as 𝑄cos𝜃1𝑋=||𝑇𝑄1𝑋||||𝜙𝑄1𝑋||.(3.16) From (3.15) and (3.16), we arrive at cos2𝜃𝑄1𝑋=𝑄𝑔1𝑄𝑋,1𝑇2𝑄1𝑋||𝑄1𝑋||2.(3.17) Taking into account that 𝜃(𝑄1𝑋) is constant on 𝐷𝜃, from (3.17) we infer that 𝑄1𝑇2𝑄1𝑋=𝜆𝑄1𝑋=𝜆𝑄1𝑋𝜂𝑄1𝑋𝑉,(3.18) where we have used 𝜆=cos2𝜃(𝑄1𝑋). Clearly 𝜆[1,0]. Converse part directly follows from (i) and (ii).

Theorem 3.5. The necessary and sufficient condition for a 2𝑞-lightlike submanifold 𝑀 of an indefinite Kenmotsu manifold 𝑀 of index 2𝑞 with structure vector field 𝑉 tangent to 𝑀 to be hemi-slant lightlike submanifold is that (i)𝜙(ltr(𝑇𝑀)) is a distribution on 𝑀, (ii)for any vector field X tangent to 𝑀 linearly independent of structure vector field 𝑉, there exists a constant 𝜇[1,0] such that 𝐵𝐹𝑄1𝑋=𝜇𝑄1𝑋𝜂𝑄1𝑋𝑉,(3.19)
where 𝜆=cos2𝜃 and 𝜃 is the slant angle of 𝑀.

Proof. If 𝑀 is a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2𝑞, then 𝜙(ltr(𝑇𝑀)) is a distribution on 𝑀. For the proof of (ii), applying 𝜙 to (3.12) and using (3.9) and (3.10), we arrive at 𝑄𝑋+𝜂(𝑋)𝑉=1𝑇2𝑄1𝑋+𝐹𝑄1𝑇𝑄1𝑋+𝐵𝐹𝑄1𝑋+𝐶𝐹𝑄1𝑋+𝐵𝐹𝑄2𝑋.(3.20) Comparing the components of screen distribution on both sides of the above equation, we get 𝑄1𝑋+𝜂𝑄1𝑋𝑄𝑉=1𝑇2𝑄1𝑋+𝐵𝐹𝑄1𝑋.(3.21) In view of Theorem 3.4 and the fact that 𝑀 is a hemi-slant lightlike submanifold, we conclude that 𝑄1𝑇2𝑄1𝑋=cos2𝜃𝑄1𝑋𝜂𝑄1𝑋𝑉.(3.22) Hence (ii) follows from (3.21) and (3.22) together with 𝐹𝑉=0. Conversely, assume that (i) and (ii) hold. From (ii) and (3.21), we observe that 𝑄1𝑇2𝑄1𝑋=(1+𝜇)𝑄1𝑋.(3.23) If we take (1+𝜇)=𝜆, then 𝜆[1,0]. Hence the proof of Theorem follows from Theorem 3.4.

Corollary 3.6. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑔𝑇𝑄1𝑋,𝑇𝑄1𝑌=cos2𝜃||𝑆(𝑇𝑀)𝑔𝑄1𝑋,𝑄1𝑌𝜂𝑄1𝑋𝜂𝑄1𝑌𝑔𝐹,(3.24)𝑄1𝑋,𝐹𝑄1𝑌=sin2𝜃||𝑆(𝑇𝑀)𝑔𝑄1𝑋,𝑄1𝑌𝜂𝑄1𝑋𝜂𝑄1𝑌(3.25) for any 𝑋,𝑌Γ(𝑇𝑀).

Differentiating (3.12), taking into account that 𝑀 is an indefinite Kenmotsu manifold and then comparing tangential, lightlike transversal and screen transversal parts, we obtain 𝑋𝑇𝑄1𝑌=𝐴𝐹𝑄1𝑌𝑋+𝐴𝐹𝑄2𝑌𝑋+𝐵𝑙(𝑋,𝑌)+𝐵𝑠(𝑋,𝑌)𝑔(𝜙𝑋,𝑌)𝑉+𝜂(𝑌)𝑇𝑄1𝑋(3.26)𝑋𝐹𝑄2𝑌=𝑙𝑋,𝑇𝑄1𝑌𝐷𝑙𝑋,𝐹𝑄1𝑌+𝜂(𝑌)𝐹𝑄2𝑋(3.27)𝑋𝐹𝑄1𝑌=𝐶𝑠(𝑋,𝑌)𝑠𝑋,𝑇𝑄1𝑌𝐷𝑠𝑋,𝐹𝑄2𝑌+𝜂(𝑌)𝐹𝑄1𝑋.(3.28)

The integrability conditions of the distributions involved in the definition of hemi-slant lightlike submanifolds are given by the following theorems.

Theorem 3.7. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2𝑞. Then the distribution Rad𝑇𝑀 is integrable if and only if(i)𝐴𝐹𝑄2𝑌𝑋𝐴𝐹𝑄2𝑋𝑌=0;(ii)𝐷𝑠(𝑋,𝐹𝑄2𝑌)=𝐷𝑠(𝑌,𝐹𝑄2𝑋)for any 𝑋,𝑌Γ(Rad𝑇𝑀).

Proof. For any 𝑋,𝑌Γ(Rad𝑇𝑀), from (3.26) we have 𝑇𝑄1𝑋𝑌=𝐴𝐹𝑄2𝑌𝑋+𝐵𝑙(𝑋,𝑌)+𝐵𝑠(𝑋,𝑌)𝑔(𝜙𝑋,𝑌)𝑉.(3.29) Interchanging 𝑋 and 𝑌 in (3.29) and subtracting (3.29) from the resulting equation, we get 𝑇𝑄1[]𝑋,𝑌=𝐴𝐹𝑄2𝑋𝑌𝐴𝐹𝑄2𝑌𝑋+2𝑔(𝜙𝑋,𝑌)𝑉.(3.30) Taking inner product of the above equation with 𝜙𝑍 for 𝑍Γ(𝐷𝜃), after simplification, we obtain []𝑔(𝑋,𝑌,𝑍)=sec2𝜃𝑔𝐴𝐹𝑄2𝑋𝑌𝐴𝐹𝑄2𝑌𝑋.,𝜙𝑍(3.31) Also, from (3.28) we have 𝐹𝑄1𝑋𝑌=𝐶𝑠(𝑋,𝑌)𝐷𝑠𝑋,𝐹𝑄2𝑌.(3.32) Interchanging the role of 𝑋 and 𝑌 in (3.32) and subtracting (3.32) from the resulting equation, we get 𝐹𝑄1[]𝑋,𝑌=𝐷𝑠𝑋,𝐹𝑄2𝑌𝐷𝑠𝑌,𝐹𝑄2𝑋.(3.33) Taking inner product of above equation with 𝜙𝑍, we arrive at []𝑔(𝑋,𝑌,𝑍)=cosec2𝐷𝜃𝑔𝑠𝑋,𝐹𝑄2𝑌𝐷𝑠𝑌,𝐹𝑄2𝑋.,𝜙𝑍(3.34) Hence our assertion follows from (3.31) and (3.34).

Theorem 3.8. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2𝑞. Then the distribution 𝐷𝜃{𝑉} is integrable if and only if 𝑙𝑋,𝑇𝑄1𝑌𝑙𝑌,𝑇𝑄1𝑋+𝐷𝑙𝑋,𝐹𝑄1𝑌𝐷𝑙𝑌,𝐹𝑄1𝑋=0,(3.35) for any 𝑋,𝑌Γ(𝑆(𝑇𝑀)).

Proof. For any 𝑋,𝑌Γ(𝐷𝜃{𝑉}), from (3.27) we have 𝐹𝑄2𝑋𝑌=𝑙𝑋,𝑇𝑄1Y𝐷𝑙𝑋,𝐹𝑄1𝑌.(3.36) Interchanging 𝑋 and 𝑌 in (3.36) and subtracting (3.36) from the resulting equation, we obtain 𝐹𝑄2[]𝑋,𝑌=𝑙𝑋,𝑇𝑄1𝑌𝑙𝑌,𝑇𝑄1𝑋+𝐷𝑙𝑋,𝐹𝑄1𝑌𝐷𝑙𝑌,𝐹𝑄1𝑋,(3.37) which proves our assertion.

In view of the above argument, if we restrict the vector fields 𝑋 and 𝑌 to 𝐷𝜃, then we see that the distribution 𝐷𝜃 is not integrable.

The following two theorems deals with the totally geodesic foliations of the leaves of the distributions 𝑆(𝑇𝑀)=𝐷𝜃{𝑉} and Rad𝑇𝑀.

Theorem 3.9. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then the distribution 𝐷𝜃{𝑉} defines totally geodesic foliation in 𝑀 if and only if 𝐴𝐹𝑄1𝑇𝑄1𝑌𝑋𝐵𝐷𝑙(𝑋,𝐹𝑄1Y) has no component in Rad𝑇𝑀 for any 𝑋,𝑌Γ(𝑆(𝑇𝑀)).

Proof. Using (2.7), (2.9), (3.12), and (3.14), for any 𝑋,𝑌Γ(𝑆(𝑇𝑀)) and 𝑁Γ(ltr(𝑇𝑀)), we obtain 𝑔𝑋𝑌,𝑁=cos2𝑔𝑋+𝑌,𝑁𝑔𝐴𝐹𝑄1𝑇𝑄1𝑌𝑋𝐵𝐷𝑙𝑋,𝐹𝑄1𝑌,𝑁,(3.38) which can be written as sin2𝜃𝑔𝑋=𝑌,𝑁𝑔𝐴𝐹𝑄1𝑇𝑄1𝑌𝑋𝐵𝐷𝑙𝑋,𝐹𝑄1𝑌,𝑁.(3.39) Thus, our assertion follows from (3.39).

Theorem 3.10. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2𝑞. Then the distribution Rad𝑇𝑀 defines totally geodesic foliation in 𝑀 if and only if (𝜉1,𝑇𝑄1𝑋)𝐴𝐹𝑄1𝑋𝜉1 has no component in Rad𝑇𝑀 for any 𝜉1Γ(Rad𝑇𝑀) and 𝑋Γ(𝐷𝜃).

Proof. For any 𝜉1,𝜉2Γ(Rad𝑇𝑀) and 𝑋Γ(𝐷𝜃), from (2.7), (2.9), (2.12), and (3.12), we obtain 𝑔𝜉1𝜉2,𝑋=𝑔𝜙𝜉2,𝜉1,𝑇𝑄1𝑋𝐴𝐹𝑄1𝑋𝜉1.(3.40) Hence our assertion follows from (3.40).

The necessary and sufficient conditions under which the induced connection on a hemi-slant lightlike submanifold immersed in indefinite Kenmotsu manifold to be metric connection is given by the following result.

Theorem 3.11. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifolds 𝑀. Then the induced connection on 𝑀 is a metric connection if and only if 𝑇𝑄1𝐴𝐹𝑄2𝑌𝑋𝐵𝐷𝑠𝑋,𝐹𝑄2𝑌=0,(3.41) for any 𝑋Γ(𝑇𝑀) and 𝑌Γ(Rad𝑇𝑀).

Proof. For 𝑋Γ(𝑇𝑀) and 𝑌Γ(Rad𝑇𝑀), from (3.1), we have 𝑋𝑌=𝜙𝑋𝜙𝑌𝑋𝜙𝜙𝑌.(3.42) Making use of (2.3), (2.7), and (2.9) in the above equation, we arrive at 𝑋𝑌+𝑙(𝑋,𝑌)+𝑠(𝑋,𝑌)=𝜙𝐴𝐹𝑄2𝑌𝑋+𝑙𝑋𝐹𝑄2𝑌+𝐷𝑠𝑋,𝐹𝑄2𝑌+𝑔(𝜙𝑋,𝜙𝑌)𝑉.(3.43) Using (3.9) and (3.10) in (3.43) and comparing the tangential components of the resulting equation, we obtain 𝑋𝑌=𝑇𝑄1𝐴𝐹𝑄2𝑌𝑋𝐵𝑙𝑋𝐹𝑄2𝑌𝐵𝐷𝑠𝑋,𝐹𝑄2𝑌,(3.44) from which our assertion follows.

Theorem 3.12. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑇 is never parallel on 𝑀.

Proof. For 𝑋Γ(𝑇𝑀), from (2.2) and (2.7), we have 𝑋𝑉+𝑙(𝑋,𝑉)+𝑠(𝑋,𝑉)=𝑋+𝜂(𝑋)𝑉.(3.45) Comparing the tangential Components of the above equation, we get 𝑋𝑉=𝑋+𝜂(𝑋)𝑉.(3.46) Since 𝑇𝑉=0, so differentiating 𝑇𝑉=0 covariantly with respect to 𝑋 we obtain 𝑋𝑇𝑉+𝑇𝑋𝑉=0.(3.47) Using (3.46) in the above equation, we get 𝑋𝑇𝑉=𝑇𝑋,(3.48) from which our assertion follows.

Similar to the notion of curvature invariant submanifolds given by Atceken and Kilic [13], we define curvature invariant lightlike submanifolds as follows.

Definition 3.13. Let 𝑀 be a 2𝑞-lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is said to be curvature invariant lightlike submanifold of 𝑀 if the covariant derivatives of the second fundamental forms 𝑙 and 𝑠 of 𝑀 satisfy 𝑋𝑙(𝑌,𝑍)𝑌𝑙(𝑋,Z)=0,𝑋𝑠(𝑌,𝑍)𝑌𝑠(𝑋,𝑍)=0,(3.49) for any 𝑋,𝑌Γ(𝑇𝑀).

Theorem 3.14. There does not exist any curvature-invariant proper hemi-slant lightlike submanifold of an indefinite Kenmotsu space form M(𝑐) with 𝑐1.

Proof. Using (2.15) and (2.16), for any 𝑋,𝑌Γ(𝑆(𝑇𝑀)) and 𝑍Γ(Rad𝑇𝑀), we obtain 𝑋𝑠(𝑌,𝑍)𝑌𝑠(𝑋,𝑍)=0,(3.50)𝑋𝑙(𝑌,𝑍)𝑌𝑙(𝑋,𝑍)=𝑐+12𝑔𝑇𝑄1𝑋,𝑌𝐹𝑄2𝑍.(3.51) Thus, our assertion follows from (3.50) and (3.51).

4. Minimal Hemi-Slant Submanifolds

In this section, we study minimal hemi-slant lightlike submanifolds immersed in an indefinite Kenmotsu manifold. First we recall the following definition of minimal lightlike submanifolds of a semi-Riemannian manifold 𝑀 given by Bejan and Duggal [14].

Definition 4.1. A lightlike submanifold (𝑀,𝑔,𝑆(𝑇𝑀)) isometrically immersed in a semi-Riemannian manifold (𝑀,𝑔) is said to be minimal if(i)𝑠=0 on Rad𝑇𝑀 and(ii)trace =0,
where trace is written with respect to 𝑔 restricted to 𝑆(𝑇𝑀).

In [14], it has been shown that the above definition is independent of 𝑆(T𝑀) and 𝑆(𝑇𝑀), but it depends on the choice of transversal bundle tr(𝑇𝑀).

For the existence of proper minimal hemi-slant lightlike submanifolds, we give the following example.

Example 4.2. Let 𝑀=(𝑅92,𝑔) be a semi-Euclidean space of signature (,,+,+,+,+,+,+,+) with respect to canonical basis (𝜕𝑥1,𝜕𝑦1,𝜕𝑥2,𝜕𝑦2,𝜕𝑥3,𝜕𝑦3,𝜕𝑥4,𝜕𝑦4,𝜕𝑧).
Consider an almost contact structure 𝜙 on 𝑅92 defined by 𝜙(𝑥1, 𝑦1, 𝑥2, 𝑦2, 𝑥3, 𝑦3, 𝑥4, 𝑦4, 𝑧)=(𝑦1,𝑥1, 𝑦2,𝑥2, 𝑥4cos𝛼𝑦3sin𝛼, 𝑦4cos𝛼+𝑥3sin𝛼, 𝑥3cos𝛼+𝑦4sin𝛼, 𝑦3cos𝛼𝑥4sin𝛼,0).
For some 𝛼(0,𝜋/2). Let 𝑀 be a submanifold of (𝑅92,𝜙) defined by the following equations: 𝑥1=𝑢1,𝑥2=𝑢2,𝑥3=𝑢3,𝑥4=𝑢4,𝑦1=𝑢2,𝑦2=𝑢1,𝑦3=sin𝑢3sinh𝑢4,𝑦4=cos𝑢3cosh𝑢4.(4.1) Then the tangent bundle 𝑇𝑀 of 𝑀 is spanned by 𝜉1=𝑒𝑧𝜕𝑥1+𝜕𝑦2,𝜉2=𝑒𝑧𝜕𝑥2+𝜕𝑦1,𝑋1=𝑒𝑧𝜕𝑥3+cos𝑢3sinh𝑢4𝜕𝑦3sin𝑢3cosh𝑢4𝜕𝑦4,𝑋2=𝑒𝑧𝜕𝑥4+sin𝑢3cosh𝑢4𝜕𝑦3+cos𝑢3sinh𝑢4𝜕𝑦4,𝑉=𝜕𝑧.(4.2) Suppose that the distribution Rad𝑇𝑀=span{𝜉1,𝜉2}. Then 𝑀 is a 2-lightlike submanifold of 𝑀. If we take 𝑆(𝑇𝑀)=span{𝑋1,𝑋2{𝑉}}, then 𝑆(𝑇𝑀) is Riemannian vector subbundle and one can easily see that 𝑆(𝑇𝑀) is a slant distribution with slant angle 𝛼. A direct calculation shows that the screen transversal bundle is spanned by 𝑊1=𝑒𝑧cosh𝑢4sinh𝑢4𝜕𝑥3+cos𝑢3cosh𝑢4𝜕𝑦3sin𝑢3cos𝑢3𝜕𝑥4+sin𝑢3sinh𝑢4𝜕𝑦4,𝑊2=𝑒𝑧cos𝑢3sin𝑢3𝜕𝑥3+sin𝑢3sinh𝑢4𝜕𝑦3cosh𝑢4sinh𝑢4𝜕𝑥4+cos𝑢3cosh𝑢4𝜕𝑦4,(4.3) and ltr(𝑇𝑀) is spanned by 𝑁1𝑒=𝑧2𝜕𝑦2+𝜕𝑥1,𝑁2=𝑒𝑧2𝜕𝑦1+𝜕𝑥2.(4.4) For a vector field 𝑋 tangent to 𝑀, from Gauss formula and after calculations, we get 𝑙=0,𝑠𝑋,𝜉1=0,𝑠𝑋,𝜉2=0,𝑠𝑋1,𝑋1=𝑒𝑧sinh2𝑢4+cos2𝑢3cosh4𝑢4sin4𝑢3𝑊2,𝑠𝑋2,𝑋2=𝑒𝑧sinh2𝑢4+cos2𝑢3cosh4𝑢4sin4𝑢3𝑊2,𝑠𝑋1,𝑋2=𝑒𝑧sinh2𝑢4+cos2𝑢3cosh4𝑢4sin4𝑢3𝑊1.(4.5) Hence, 𝑀 is a minimal proper hemi-slant lightlike submanifold of 𝑅92, and the induced connection on 𝑀 is a metric connection. We note that 𝑀 is neither totally geodesic nor totally umbilical.

Now, we prove a lemma which will be helpful for the study of minimal hemi-slant lightlike submanifolds.

Lemma 4.3. Let 𝑀 be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 such that dim(𝐷𝜃)=dim(𝑆(𝑇𝑀)). If {𝑒1,𝑒2,,𝑒𝑚} is a local orthonormal basis of Γ(𝐷𝜃), then {csc𝜃𝐹𝑒1,,csc𝜃𝐹𝑒𝑚} is an orthonormal basis of 𝑆(𝑇𝑀).

Proof. Suppose that {𝑒1,𝑒2,,𝑒𝑚} is a local orthonormal basis of 𝐷𝜃. From (3.25) together with the fact that 𝐷𝜃 is Riemannian, we conclude that 𝑔csc𝜃𝐹𝑒𝑖,csc𝜃𝐹𝑒𝑗=𝛿𝑖𝑗,(4.6) where 𝑖,𝑗=1,2,3,,𝑚. Hence our assertion follows from (4.6).

The existence of minimal hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold is given by the following two theorems.

Theorem 4.4. Let 𝑀 be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is minimal if and only if trace𝐴𝜉𝑗|||𝑆(𝑇𝑀)=0,trace𝐴𝑊𝛼||𝑆(𝑇𝑀)𝑔𝐷=0,𝑙(𝑋,𝑊),𝑌=0,(4.7) for any 𝑋,𝑌Γ(Rad𝑇𝑀) and 𝑊Γ(𝑆(𝑇M)), where {𝜉𝑗}𝑟𝑗=1 is a basis of Rad𝑇𝑀 and {𝑊𝛼}𝑚𝛼=1 is a basis of 𝑆(𝑇𝑀).

Proof. Using 𝑉𝑉=0 and (2.2), we obtain 𝑙(𝑉,𝑉)=𝑠(𝑉,𝑉)=0. It is known that 𝑙=0 on Rad𝑇𝑀 [14, Proposition 4.1]. If we take {𝑒1,𝑒2,,𝑒𝑚} as an orthonormal basis of 𝐷𝜃, then 𝑀 is minimal if and only if Σ𝑚𝑘=1𝑒𝑘,𝑒𝑘=0(4.8) and 𝑠=0 on Rad𝑇𝑀. From (2.10) and (2.14), we have Σ𝑚𝑘=1𝑒𝑘,𝑒𝑘=Σ𝑚𝑘=11𝑟Σ𝑚𝑗=1𝑔𝐴𝜉𝑗𝑒𝑘,𝑒𝑘𝑁𝑗+1𝑚Σ𝑚𝛼=1𝑔𝐴𝑊𝛼𝑒𝑘,𝑒𝑘𝑊𝛼.(4.9) On the other hand, from (2.10), we infer that 𝑠=0 on Rad𝑇𝑀 if 𝑔𝐷𝑙(𝑋,𝑊),𝑌=0,(4.10) for 𝑋,𝑌Γ(Rad𝑇𝑀) and 𝑊Γ(𝑆(𝑇𝑀)). Thus, our assertion follows from (4.9) and (4.10).

Theorem 4.5. Let 𝑀 be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 such that dim(𝐷𝜃)=dim(𝑆(𝑇𝑀)). Then 𝑀 is minimal if and only if trace𝐴𝜉𝑗|||𝑆(𝑇𝑀)=0,trace𝐴𝑊𝑒𝑖||𝑆(𝑇𝑀)𝑔𝐷=0,𝑙X,𝐹𝑒𝑖,𝑌=0.(4.11) for 𝑋,𝑌Γ(Rad𝑇𝑀), where {𝜉𝑘}𝑟𝑘=1 is a basis of Rad𝑇𝑀 and {𝑒𝑗}𝑚𝑗=1 is a basis of 𝐷𝜃.

Proof. From 𝑉𝑉=0 and (2.2), we obtain 𝑙(𝑉,𝑉)=𝑠(𝑉,𝑉)=0. Recall that 𝑙=0 on Rad𝑇𝑀 [14, Proposition 4.1]. In view of Lemma 4.3, we observe that {csc𝜃𝐹𝑒1,,csc𝜃𝐹𝑒𝑚} is an orthonormal basis of 𝑆(𝑇𝑀). Then, for any 𝑋Γ(𝑇𝑀) and for some function 𝐴𝑖, 𝑖(1,2,,𝑚), we can write 𝑠(𝑋,𝑋)=Σ𝑚𝑖=1𝐴𝑖csc𝜃𝐹𝑒𝑖,(4.12) from which we have 𝑠(𝑋,𝑋)=Σ𝑚𝑖=1𝐴csc𝜃𝑔𝐹𝑒𝑖𝑋,𝑋𝐹𝑒𝑖,(4.13) for any 𝑋Γ(𝑆(𝑇𝑀)). Thus our assertion follows from the above equation and Theorem 4.4.

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